Null controllability with vanishing energy for discrete-time systems in Hilbert space

In this paper null controllability with vanishing energy is considered for discrete-time systems in Hilbert space. As in the case of continuous time systems necessary and sufficient conditions in terms of an algebraic Riccati equation are given. Then necessary and sufficient conditions involving the spectrum of the system operator are given. Reachability and controllability with vanishing energy are also considered, and necessary and sufficient conditions for them are given. Finally applications to sampled-data systems, systems with impulse control and periodic systems are discussed.